
where \({\in _X} \subseteq X\times \mathcal {Q} X\)  is the element-of relation for \(X\) .

As shown by Marti and Venema [1]}, the lax nabla is in fact
a natural transformation \(\nabla ^L: T\mathcal {Q} \Rightarrow \mathcal {Q} T^\mathsf {op}\) ,
and coincides with Moss' classical \(\nabla \)  for \(L\)  being the Barr
extension of \(T\)  (and \(T\)  preserving weak pullbacks). In combination
with a functor presentation, the lax nabla gives rise to a family of
predicate liftings:
